Problem: Simplify the following expression: $ z = \dfrac{1}{3} - \dfrac{-3k - 3}{k - 2} $
Explanation: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{k - 2}{k - 2}$ $ \dfrac{1}{3} \times \dfrac{k - 2}{k - 2} = \dfrac{k - 2}{3k - 6} $ Multiply the second expression by $\dfrac{3}{3}$ $ \dfrac{-3k - 3}{k - 2} \times \dfrac{3}{3} = \dfrac{-9k - 9}{3k - 6} $ Therefore $ z = \dfrac{k - 2}{3k - 6} - \dfrac{-9k - 9}{3k - 6} $ Now the expressions have the same denominator we can simply subtract the numerators: $z = \dfrac{k - 2 - (-9k - 9) }{3k - 6} $ Distribute the negative sign: $z = \dfrac{k - 2 + 9k + 9}{3k - 6}$ $z = \dfrac{10k + 7}{3k - 6}$